Effective speed of sound in phononic crystals A.A. Kutsenko a , A.L. Shuva lov a , A.N. Nor ris a,b a Univers it´ e de Bordeaux, Institut de M´ ecanique et d’Ing´ enieri e de Bordeaux, UMR 5295, Talence 33405, F rance, b Mechanic al and Aer ospace Engineering, Rut gers University , Pisc atawa y, NJ 08854 , USA A new formula for the effective quasistatic speed of soundc in 2D and 3D periodic materials is report ed. The approac h uses a monodro my- matr ix operator to enabl e direc t int egration in one ofthe coordinat es and exponentially fast convergence in other s. As a resu lt, the solution forc has a more closed form than prev ious formulas. It significan tly improv es the efficie ncy and accura cy ofevaluating c for high-contrast composites as demonstrated by a 2D example with extreme behavior. P ACS numbers: 62.65.+k, 43.20.+g, 02.70.Hm, 43.90.+v I. INTRODUCTION Long- stand ing interest in modelling effec tive elastic propert ies of compos ites with micro struc ture has sub- stantially intensified with the emerging possibility of de- signing periodic structures in air 1 and in solids 2 to form phononic crystals and other exotic metamaterials, which open up exciting application prospects ranging from neg- ative index lenses to small sca le multiband pho non ic devices 3 . This new prospect ive bring s about the need for fast and accurate computational schemes to test ideas insilico. The most common numer ical tool is the F ourier or plane- wa ve expansion method (PWE). It is widely used for calculating various spectral parameters includ- ing the effective quasistatic speed of sound in acoustic 4 and elastic 5 pho nonic cryst als . At the same time, the PWE calculation is known to face problems when ap- plied to high-contrast composites 3 , which are of especial interest for applications. Particularly riveting is the case where a soft ingredient is embedded in a way breaking the connectivity of densely packed regions of stiff ingre- dient. Phy sically speaking, the speed of sound, which is large in a homogeneously stiff medium, should fall dra- matically when even a small amount of soft component forms a ’quasi-insulating network’. Note that this case, which implies a strong effect of multiple interactions, is also ungainly for the multiple-scattering approach 1,2 . The purpose of present Letter is to highlight a new method for ev aluating the quasistat ic effective sou nd speed c in 2D and 3D phononic cr ys ta ls. The idea is to recast the wave equation as a 1st-order ’ordinary’ dif- ferential system (ODS) with respect to one coordinate (say x 1 ) and to use a monodromy-matrix operator de- fined as a multiplicative (or path) integral in x 1 . By thi s means, we derive a for mula for c whose essential advantages are an explicit integration in x 1 and an expo- nentially small error of truncation in other coordinate(s). Both these features of the analytical result are shown to significantly improve the efficiency and accuracy of its numerical implementation in comparison with the con- ven tional PWE calcu lation , whic h is demo nstrated for a 2D steel /epox y square lattic e. The powe r of the new approach is especially apparent at high concentration fof steel inclusions, where the effective speed c displays a steep, near vertical, dependence for f≈1, a feature not captured by conventional techniques like PWE. II. EFFECTIVE SPEED: 2D ACOUSTIC W AVES A. SETUP. Consider the scalar wave equation ∇ · (µ∇ ∇ ∇v) = −ρω 2 v, (1) for time-harmonic shear displacementv (x, t) = v (x)e −iωt in a 2D solid continuum 8 with T-periodic density ρ(x) and shear coefficient µ(x). Assume a square unit cell T= i t i a i = [0, 1] 2 with unit translation vectors a 1 ⊥ a 2 taken as the basis for x = i x i a i . Imposing the Floquet condition v(x) = u(x)e ik·x where u(x) is periodic and k=kκ κ κ ( | κ κ| = 1), Eq. (1) becomes (C0 + C1 + C2 )u= ρω 2 u with C0 u= −∇(µ∇ ∇u), C1 u= −ik · (µ∇ ∇u+ ∇(µu)), C2 u=k 2 µu. (2) Regular perturbation theory applied to (2) yields the ef- fective speed c (κ κ) = lim ω,k→0 ω(k)/k in the form 6 c 2 (κ κ) = µ eff ( κ κ)/ρ, µ eff (κ κ κ) = µ− M( κ κ) with (3) M(κ κ) = 2 i,j=1 Mij κ i κ j , Mij = C−1 0 ∂ i µ, ∂ j µ =Mji , where ∂ i ≡ ∂/∂x i , spatial averages are defined by f≡ T f(x)dx =f1 2 , fi ≡ 1 0 f(x)dx i , (4) and (·, ·) denotes the scalar product in L 2 (T) so that (f, h) =f h ∗ ( ∗ mean s comp lex conjugat ion). The dif- ficulty with (3) is that it involves the inverse of a par- tial differ ent ial operat or C0 . One solution is to apply a double Fourier expansion to C−1 0 and ∂ i µ in (3). Th is leads to the PWE formula for the effective speed 4 which is expressed via infinite vectors and the inverse of the infinite matrix of Fourier coefficients ofµ(x). Numeri - cal implementation of the PWE formula requires dealing with large dense matrices, especially in the case of high- contrast composites for which the PWE convergence is slow (see§ IV). An alternative ”brute force” procedure ofa r X i v : 1 1 0 6 . 5 4 1 2 v 1 [ m a t h p h ] 2 7 J u n 2 0 1 1
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